Interval estimation of nonlinear parametric regression in a passive experiment

The paper presents a method for constructing point and interval estimates of the regression coefficient (RC) of nonlinear regression in a passive, in a certain sense, experiment. Passivity is understood through the role of the experimenter regarding the content of the initial data. The role is passive and does not affect the method of collecting information. The initial data for the experimenter is an unchangeable given. The initial evaluation data are the observed realizations of RC (RRC), which are the result of solving the inverse problem: finding the RC parameter of the regression function from the measured values of its dependent and independent variables without taking into account errors. A multilevel point estimate is constructed as the average of sequentially nested subsets of the set of RRCs, which are formed based on approximations of numerical characteristics (mathematical expectation and variance) RRC of varying degrees of accuracy. Approximation formulas were found earlier. The accuracy of the point estimation increases simultaneously with the strictness of the conditions for the possibility of approximations. At the same time, for the roughest and most accurate estimates, the approximations of the variances coincide with the accuracy of the approximation, which is a sign of non-randomness of the estimates. They are a classical approximation of known accuracy. As a result, it becomes possible to construct a reliable interval covering the true value of the RC. The paper presents the results of a numerical (simulation) experiment of two nonlinear relative to the estimated parameter and an independent regression variable. One experiment differs from another: 1) the model value of the RC, 2) the intervals of setting the area of definition of the function (its independent variable), 3) the magnitude of the standard deviation of the random component of the model, 4) the amount of initial data. As a result, the properties of the functions change. In some experiments, the values of derived functions for the estimated parameter are greater than one, which is a sign of significant nonlinearity of the function, in others they are less than one or close to zero, which is a sign of the internal linearity of the function relative to the estimated parameter. The evaluation results are consistent with the theoretical justifications of the assessments. If the conditions for the formation of subsets of the RRC set are met, the true values of the RC are reliably covered by an interval determined by the first and third degrees of the standard deviation of the random component of the model. With the increase in the number of initial evaluation data, the accuracy of the result increases, which is consistent with the previously proven consistency of the estimates. In a passive experiment, it is possible to optimize point and interval estimation based on the average sample characteristics of the initial evaluation data. The evaluation of the RC is carried out without relying on the distribution law of the random component of the model, which refers the evaluation method to the "robust" method.